The advantge of the topological methods is that they require relatively little information regarding the dynamics on the normally hyperbolic invariant cylinder. In particular, no knowlege of invariant quasiperiodic tori is expected. As far as the authors know, correctly aligned windows were imagined by Conley and Easton [1]}, and the first application of the correctly aligned windows method to Arnold diffusion problems is the paper [2]}. In this paper, Gidea and de la Llave use the method of correctly aligned windows to prove the existence of diffusion orbits in a priori unstable Hamiltonian systems and to construct orbits with an unbounded growth of energy for the Mather problem (that is a geodesic flow with a time dependent potential). This second model is usually said to be a priori chaotic. In both settings the normally hyperbolic invariant cylinder is two-dimensional. Moreover, they assume that the induced dynamics on the cylinder satisfies a twist property, and the twist is uniform with respect to the perturbative parameter.
